Letters for the sets of rational and real numbers.
The authors of classical textbooks such as Weber and Fricke did not denote particular
domains of computation with letters.

In 1872 Richard Dedekind denoted the rationals by R and
the reals by blackletter R in Stetigkeit und irrationale Zahlen (1872) (Continuity and irrational numbersWorks,3, 315-334.
Dedekind also used K for the integers and J for complex numbers.

In 1888 Richard Dedekind denoted the natural numbers by N in Was ist und was sollen die Zahlen, §6.

In 1889 Giuseppe Peano cited Dedekind’s book in his Arithmetices prinicipia nova methodo exposita, and used the same symbol for the positive integers as Dedekind. Peano used N, R, and Q, and showed their meaning in a table on page 23:

N

numerus integer positivus

R

num. rationalis positivus

Q

quantitas, sive numerus realis positivus

In 1895 in his Formulaire de mathématiques,
Peano used N for the positive integers, n
for integers, N0 for the positive integers and
zero, R for positive rational numbers, r for rational
numbers, Q for positive real numbers, q for real
numbers, and Q0 for positive real numbers and zero
[Cajori vol. 2, page 299].

In 1897 Peano used N1 instead of N. [Wilfried Neumaier]

In 1926 Helmut Hasse (1898-1979) used Γ for the integers and
Ρ (capital rho) for the rationals in Höhere Algebra I and
II, Berlin 1926. He kept to this notation in his later books on
number theory. Hasse's choice of gamma and rho may have been
determined by the initial letters of the German terms "ganze Zahl"
(integer) and "rationale Zahl" (rational).

In 1929 Otto Haupt used G0 for the integers and
Ρ0 (capital rho) for the rationals in Einführung in die
Algebra I and II, Leipzig 1929.

In 1930 Bartel Leendert van der Waerden used C for the integers
and Γ for the rationals in Moderne Algebra I,
Berlin 1930, but in editions during the sixties, he changed to Z and
Q.

In 1930 Edmund Landau denoted the set of integers by a fraktur Z
with a bar over it in Grundlagen der Analysis (1930, p. 64).
He does not seem to introduce symbols for the sets of rationals,
reals, or complex numbers.

Q for the set of rational numbers and Z for the set of
integers are apparently due to N. Bourbaki. (N. Bourbaki was a group
of mostly French mathematicians which began meeting in the 1930s,
aiming to write a thorough unified account of all mathematics.) The
letters stand for the German Quotient and Zahlen. These
notations occur in Bourbaki's Algébre, Chapter 1.

Julio González Cabillón writes that he believes
Bourbaki was responsible for both of the above symbols, quoting Weil,
who wrote, "...it was high time to fix these notations once and for
all, and indeed the ones we proposed, which introduced a number of
modifications to the notations previously in use, met with general
approval."

C for the set of complex numbers. William C. Waterhouse
wrote to a history of mathematics mailing list in 2001:

Checking things I have available, I found C used for the
complex numbers in an early paper by Nathan Jacobson:

Structure and Automorphisms of Semi-Simple Lie Groups in
the Large, Annals of Math. 40 (1939), 755-763.

The second edition of Birkhoff and MacLane, Survey of Modern
Algebra (1953), also uses C (but is not using the Bourbaki
system: it has J for integers, R for rationals, R^# for reals). I
have't seen the first edition (1941), but I would expect to find C
used there too. I'm sure I remember C used in this sense in a number
of other American books published around 1950.

I think the first Bourbaki volume published was the results summary
on set theory, in 1939, and it does not contain any symbol for the
complex numbers. Of course Bourbaki had probably chosen the symbols
by that time, but I think in fact the first appearance of (bold-face)
C in Bourbaki was in the formal introduction of complex numbers in
Chapter 8 of the topology book (first published in
1947).

Euler's phi function (totient function). The symbol φ(m) for the number of integers less than m
that are relatively prime to m was introduced by Carl Friedrich Gauss (1777-1855) in 1801 in his
Disquisitiones arithmeticae
articles 38, 39 (p. 30) (Cajori vol. 2, page 35, and Dickson, page 113-115).

The function was first studied by Leonhard Euler
(1707-1783), although Dickson (page 113) and Cajori (vol. 2, p. 35) say that Euler did not use a functional
notation in Novi Comm. Ac. petrop., 8, 1760-1, 74, and Comm. Arith.,
1, 274, and that Euler used πN in Acta Ac. Petrop., 4 II (or 8), 1780 (1755), 18, and Comm. Arith.,
2, 127-133. Shapiro agrees, writing: "He did not employ any symbol for
the function until 1780, when he used the notation πn."

Sylvester, who introduced the name totient for the function, seems to have believed
that Euler had used φ. He writes in 1888
(
vol. IV p. 589 of his Collected Mathematical
Papers) "I am in the habit of representing the totient of n
by the symbol τ n, τ (taken from the initial of the word it
denotes) being a less hackneyed letter than Euler's φ, which has
no claim to preference over any other letter of the Greek alphabet, but rather
the reverse." This information was taken from a post in sci.math by Robert
Israel.

Legendre symbol (quadratic reciprocity).
Adrien-Marie Legendre introduced the notation
= 1 if D is a quadratic residue of p, and
= -1 if D is a quadratic non-residue of p.

According to Hardy & Wright's An Introduction to the
Theory of Numbers, "Legendre introduced 'Legendre's symbol' in his
Essai sur la theorie des nombres, first published in 1798. See, for example,
§135 of the second edition (1808)." In the third edition on
Gallica this is on p.197.

However, according to William J. Leveque in Fundamentals of Number Theory,
"Legendre introduced his symbol in an article in 1785, and at the same time stated the
reciprocity law without using the symbol."

[Both of these citations were provided by Paul Pollack.]

Mersenne numbers. Mersenne numbers are marked
Mn by Allan Cunningham in 1911 in Mathematical
Questions and Solutions from the Educational Times (Cajori vol.
2, page 41).

The norm of a + bi. Dirichlet used N(a+bi)
for the norm a2+b2
of the complex number a+bi in Crelle's Journal Vol. XXIV
(1842) (Cajori vol. 2, page 33). See NORM on Words page

Galois field. Eliakim Hastings Moore used the symbol
GF[qn] to represent the Galois field of
order qn in 1893. The modern notation is
"Galois-field of order qn" (Julio González
Cabillón and Cajori vol. 2, page 41).

Sum of the divisors of n. Euler introduced the symbol
n in a paper published in 1750 (DSB, article:
"Euler").

In 1888, James Joseph Sylvester continued the use of Euler's notation
n (Shapiro).

Allan Cunningham used σ(N) to represent the
sum of the proper divisors of N in Proceedings of the
London Mathematical Society 35 (1902-03):

The Repetition of the Sum-Factor Operation. Abstract of an informal
communication made by Lieut.-Col. A. Cuningham, June 12th, 1902.

Let σ(N) denote the sum of the sub-factors of N (including 1, but
excluding N). It was found that, with most numbers, σnN = 1, when
the operation (σ) is repeated often enough. There is a small class for
which σnN = P (a perfect number), and then repeats; another
small class for which σnN = A, σn + 1N = B,
where A, B are amicable numbers, and then repeats (A, B alternately); another small
class for which (even when N is small, < 1000) σnN increases
beyond the practical power of calculation.

[Cajori, vol. 2, page 29, and Paul Pollack]

In 1927 Landau chose the notation S(n) (Shapiro).

L. E. Dickson used s(n) for the sum of the divisors of
n (Cajori vol. 2, page 29).

The Möbius function. Möbius' work appeared in 1832
but the µ symbol was not used.

Big-O and little-o notation. According
to Wladyslaw Narkiewicz in The Development of Prime Number Theory:

The symbols O(·) and o(·) are usually called the Landau symbols.
This name is only partially correct, since it seems that the first of them appeared
first in the second volume of P. Bachmann's treatise on number theory (Bachmann,
1894). In any case Landau (1909a, p. 883) states that he had seen it for the
first time in Bachmann's book. The symbol o(·) appears first in Landau (1909a).
Earlier this relation has been usually denoted by {·}.